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Advances in Mathematical Physics
Volume 2017 (2017), Article ID 8716752, 8 pages
https://doi.org/10.1155/2017/8716752
Research Article

A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Correspondence should be addressed to Muzhou Hou; moc.anis@uohzumuoh

Received 4 June 2017; Revised 16 August 2017; Accepted 22 August 2017; Published 24 September 2017

Academic Editor: Mariano Torrisi

Copyright © 2017 Taohua Liu and Muzhou Hou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions. First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven. Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of and computational cost of . Traditionally, the Gaussian elimination method requires storage of and computational cost of . Finally, the accuracy and efficiency of the method are checked with a numerical example.